On the time reversal invariance of classical electromagnetic theory

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Studies in History and Philosophy of Modern Physics 35 (2004) 295–315

On the time reversal invariance of classical electromagnetic theory David B. Malament Department of Logic and Philosophy of Science, University of California, Irvine, CA 92697-5100, USA Received 4 July 2003; accepted 10 September 2003 This article is dedicated to the memory of Rob Clifton.

Abstract David Albert claims that classical electromagnetic theory is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are ‘‘simply wrong’’ because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I argue, to the contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory—one that makes use of the invariant four-dimensional formulation of the theory—that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. The only assumption one needs is temporal orientability. r 2004 Elsevier Ltd. All rights reserved. Keywords: Time reversal invariance; Electromagnetic theory; Relativity theory

1. Introduction In the first chapter of Time and Chance, David Albert (2000) argues that classical electromagnetic theory (in contrast, for example, to Newtonian mechanics) E-mail address: [email protected] (D.B. Malament). 1355-2198/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsb.2003.09.006

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is not time reversal invariant. He acknowledges that all physics books say that it is, but claims they are ‘‘simply wrong’’ because they rely on an incorrect account of how the time reversal operator, properly understood, acts on magnetic fields. Once that account is corrected, he believes, it is perfectly obvious that the theory is not time reversal invariant. No deep mathematics or physics is called for, only a clear understanding of the nature of the time reversal operation. Received opinion, no doubt, is often wrong. But I do not believe it is here. Physics books tell us that the time reversal operation leaves the electric field E intact, but inverts the magnetic field B: Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. He even suggests (p. 18) that the inversion of B is nothing but an ad hoc maneuver to save the time reversal invariance of classical electromagnetic theory. I will argue, to the contrary (Section 6), that the inversion of B makes good sense and is, in fact, forced by elementary geometric considerations. The argument—really just a version of one that can be found in any book on the subject—traces the asymmetric treatment of E and B to the fact that the latter, unlike the former, is not a vector field in the usual sense. (In traditional language, E is a ‘‘polar’’ vector field, while B is an ‘‘axial’’ vector field.) Before giving this response to Albert’s claims, I will make a somewhat different point. It seems to me that the inversion of magnetic fields by the time reversal operator is really something of a distraction. One can formulate and argue for the claim that classical electromagnetic theory is time reversal invariant without making reference to magnetic fields at all. I will do so in Sections 3 and 4, using the invariant four-dimensional formulation of the theory. It is my hope that the proposed way of thinking about time reversal invariance will be of interest in its own right, Albert aside. The key idea is this. The tensor fields ðFab ; J a Þ that represent the electromagnetic field and its charge-current source field are only determined relative to a choice of temporal orientation. I will construe time reversal as an operation taking pairs ðFab ; J a Þ as determined relative to one orientation to pairs ðTFab ; TJ a Þ as determined relative to the other. This approach has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. The only assumption one needs is temporal orientability. (In contrast, the standard approach presupposes that the background spacetime structure exhibits special time reflection symmetries.1) At the same time, it is fully equivalent to the standard approach when the symmetries are present, as in Minkowski spacetime.2

1 The standard approach leaves the background temporal orientation fixed, but inverts dynamical histories under the action of the symmetries. 2 My discussion of Albert’s views is closely related to those of John Earman (2002) and Frank Arntzenius (2004). They too dispute his claims about the (non) temporal invariance of classical electromagnetic theory, but offer somewhat different arguments in response.

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2. Albert’s argument I will start by presenting the standard account of time reversal invariance and then reconstructing Albert’s argument. The standard account goes something like this.3 A physical theory is said to be time reversal invariant if, for any sequence of instantaneous states SI ; y; SF allowed by the theory, the time reversed sequence R ðSF Þ; y; R ðSI Þ is allowed as well. (Here R is the time reversal operator, and the temporal order of states is understood to run from left to right.) If a time coordinate t is given, we can formulate the defining condition, somewhat more precisely, this way: for any history t/SðtÞ allowed by the theory, the time reversed history t/ðT SÞðtÞ is allowed as well, where ðTSÞðtÞ ¼ R ðSðtÞÞ:4 For these characterizations to make full sense in any particular case, of course, we have to know what count as ‘‘instantaneous states’’, and how the time reversal operator R acts on them. Consider, for example, the case of a point particle in Newtonian mechanics. Here (on the standard account) the instantaneous states are pairs ðx; vÞ; where x is the particle’s position and v its velocity; and the time reversal operator R takes the state ðx; vÞ to the state ðx; vÞ: It follows that the induced operator T takes the history t/SðtÞ ¼ ðxðtÞ; vðtÞÞ to the time reversed history t/ðT SÞðtÞ ¼ R ðxðtÞ; vðtÞÞ ¼ ðxðtÞ;  vðtÞÞ: The latter reverses the order in which the particle occupies particular positions, and inverts its velocity at every one. The latter inversion (turning v to v) makes sense. If we watch a movie of a particle moving from left to right, and then run the movie backwards, we see it moving from right to left. Since the velocity of a particle is the time derivative of its position, R must invert v:5 Next consider (the standard account of) classical electromagnetic theory. Here the instantaneous states are quadruples ðE; B; r; jÞ; where E is the electric field, B is the magnetic field, r is the charge density field, and j is the current density field. The latter two characterize the instantaneous state of the charge distribution that serves as a source for E and B: The time reversal operator R (at least according to the standard account) has the following action on these objects: ðE; B; r; jÞ

R

/

ðE;  B; r;  jÞ:

ð1Þ

3 I will, for the moment, take for granted that we have a well-defined notion of space at a given time and ignore complications involving relativity theory. 4 The notation may be confusing here. I am taking R to be an operator acting on individual instantaneous states, and taking T to be an operator acting on histories that is induced by R: The time reversed history t/ðT SÞðtÞ runs the states of t/SðtÞ in reverse temporal order and applies R to each one. 5 Since the position of the particle in the time reversed trajectory is ðT xÞðtÞ ¼ xðtÞ at time t, its velocity at t is d d ðT vÞðtÞ ¼ ðT xÞðtÞ ¼ xðtÞ ¼ vðtÞ: dt dt

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Hence the induced operator T takes the composite history t/ðEðt; xÞ; Bðt; xÞ; rðt; xÞ; jðt; xÞÞ

ð2Þ

to the time reversed history t/ððT EÞðt; xÞ; ðT BÞðt; xÞ; ðT rÞðt; xÞ; ðT jÞðt; xÞÞ;

ð3Þ

where ðT EÞðt; xÞ ¼ Eðt; xÞ;

ð4Þ

ðT BÞðt; xÞ ¼ Bðt; xÞ;

ð5Þ

ðT rÞðt; xÞ ¼ rðt; xÞ;

ð6Þ

ðT jÞðt; xÞ ¼ jðt; xÞ:

ð7Þ

These are the standard time reversal transformations. (Here I make explicit the dependence of the fields on position; Eðt; xÞ; for example, is the value of the electric field at time t and position x:) Just why (or whether) R should have the action indicated in (1)—in particular, why it should invert the magnetic field B but leave intact the electric field E—is precisely the issue I will be considering. (Albert claims it should not invert B:) But if R does act as in (1), then it follows immediately that the original history (2) satisfies Maxwell’s equations iff the time reversed history (3) does so, i.e., it follows that classical electromagnetic theory (or, at least, that fragment of the theory embodied in these equations) is time reversal invariant. This can be seen by inspection of Maxwell’s equations: r  E ¼ r; ð8Þ rE¼

@B ; @t

r  B ¼ 0; rB¼

@E þ j: @t

ð9Þ ð10Þ ð11Þ

David Albert’s objection to the standard account of time reversal invariance begins with a disagreement over how to characterize instantaneous states. Consider again the example of a particle in Newtonian mechanics. It is one of Albert’s fundamental claims that though the particle’s position at a given moment (time t0 ) deserves to be considered a component of its instantaneous state, its velocity at that moment does not. He reaches this conclusion by first insisting that (on any proper account) (i) instantaneous state descriptions at different times should be ‘‘logically, conceptually, and metaphysically independent’’ of each other (p. 10), and then pointing out that (ii) this requirement fails if we characterize instantaneous particle states in terms of positions and velocities. It fails because specification of the

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particle’s position at all times before and after t0 determines its velocity at t0 : As he puts it: What typically gets referred to in the physical literature as an ‘‘instantaneous state’’ of a Newtonian-mechanical universe, of course, is a specification of the positions and the velocities of all the particles in the world at the time in question. But the trouble with that is just that specifications of the positions and the velocities of all the particles in the world at one time are not conceptually independent of specifications of the positions and velocities of all the particles in the world at all other times. The trouble (to put it slightly differently) is that a specification of the positions and velocities of all the particles in the world at some particular instant is not a specification of the physical situation of the world at that instant alone; it is not a specification of the physical situation of the world at that instant as opposed to all others, at all! (Albert, 2000, pp. 10–11, italics in original) I am not convinced by Albert’s criticism of the standard account of instantaneous states. Responses by Earman (2002), Floyd (2003), and Smith (2003) seem to me exactly right.6 But I am not going to dwell on the criticism because it is largely irrelevant to my concerns. Albert believes that the things called ‘‘instantaneous states’’ in physics books should properly be called ‘‘dynamical conditions’’. Fine. Suppose we adopt that terminology. Then the standard characterization of time reversal invariance given above comes out as follows: a physical theory is time reversal invariant if, for any sequence of dynamical conditions SI ; y; SF allowed by the theory, the time reversed sequence R ðSF Þ; y; R ðSI Þ is allowed as well. This is not Albert’s preferred characterization7, but, as he acknowledges (p. 19), it is equivalent if one adopts what he takes to be the proper account of the action of the time reversal operator R: So I am going to put other issues (such as the nature of ‘‘instantaneous states’’) aside, and focus my attention specifically on his nonstandard account of that operator. That seems to me the heart of the matter. Albert agrees that if we have two vector quantities in a basic dynamical condition, and the second is the time derivative of the first, then if R keeps the first intact, it must reverse the second. (The argument is essentially as in footnote 5.) So he agrees that in the case of Newtonian particle mechanics, R takes the dynamical condition ðx; vÞ to ðx; vÞ: So he agrees that the theory is time reversal invariant (despite his disagreement as to whether a particle’s velocity counts as part of its ‘‘instantaneous state’’). Where he takes his stand is with the principle that R should not reverse a vector quantity in a basic dynamical condition unless it arises as the time derivative of some other quantity that R leaves intact. That is why he does not accept (1) as a 6

The papers by Floyd and Smith are directed primarily against Arntzenius (2000). But most of what they have to say applies equally well to Albert (2000). 7 He takes a physical theory to be ‘‘time reversal invariant’’ if, for any sequence of ‘‘Albert instantaneous states’’ SI ; y; SF allowed by the theory, the sequence SF ; y; SI is allowed as well. (Here ‘‘Albert instantaneous states’’ are ones that satisfy his strictures. In the case of Newtonian mechanics, they include particle positions, but not velocities.)

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characterization of R’s action in classical electromagnetic theory. Specifically, he insists that just as R leaves E intact, it should leave B intact. Magnetic fields are not the sorts of things that any proper time-reversal transformation can possibly turn around. Magnetic fields are not—either logically or conceptually—the rates of change of anything. (Albert, 2000, p. 20, italics in original) He takes the proper action of R to be, not (1), but rather8 ðE; B; r; jÞ

R ðAlbert alternativeÞ

/

ðE; B; r;  jÞ:

ð12Þ

It follows that, for him, the induced operator T takes the composite history t/ðEðt; xÞ; Bðt; xÞ; rðt; xÞ; jðt; xÞÞ to the time reversed history t/ððT EÞðt; xÞ; ðT BÞðt; xÞ; ðT rÞðt; xÞ; ðT jÞðt; xÞÞ; where, now, the terms on the right are given by the following table of Albert’s proposed time reversal transformations: ðT EÞðt; xÞ ¼ Eðt; xÞ;

ð13Þ

ðT BÞðt; xÞ ¼ Bðt; xÞ;

ð14Þ

ðTrÞðt; xÞ ¼ rðt; xÞ;

ð15Þ

ðT jÞðt; xÞ ¼ jðt; xÞ:

ð16Þ

The only difference between the standard and alternate transformations is in the treatment of B; i.e., in the difference between (5) and (14). But everything turns on that difference. If Albert is correct that the time reversal operator, properly understood, does not invert B; then classical electromagnetic theory does not qualify as time reversal invariant. (Two of Maxwell’s equations are invariant under his alternate transformations. But the other two—the second and the fourth in our list—are not.) My reconstruction of Albert’s position to this point has been rather formal. So let me also try to capture what I take to be the intuitive picture behind it. I will call it the ‘‘E and B just lie there picture’’. Consider a stacked family of simultaneity slices, with the bottom slice labeled ‘I’ (initial) and the top one labeled ‘F’ (final) (see Fig. 1). On each slice, two vectors fields, E and B are given. Each can be pictured as an array of arrows, one at each point of the slice. Consider a particle moving with uniform velocity ‘‘from left to right’’. At each point where its worldline intersects a timeslice, the E and B vectors have particular lengths and directions. Suppose that at all these 8 Albert does not discuss sources. I simply take for granted that he would have R reverse the current density j since it is a velocity-like object. But nothing turns on this assumption. I could equally well restrict attention to Maxwell’s equations in the special case where all sources vanish: r ¼ 0 ¼ j: In that case too, the equations (as a set) are invariant under the standard time reversal transformations, but not under Albert’s proposed alternative transformations.

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Fig. 1. A partial history showing the E and B fields at two points on the worldline of a particle moving from left to right.

Fig. 2. The ‘‘Albert time reversal’’ of the partial history displayed in Fig. 1. The order of the time slices I and F has been reversed, and the worldline is now that of a particle moving from right to left. But the configuration of E and B fields within each slice has been left intact.

points the vectors are non-zero, so that there is a well-defined angle between them. Let it be yI at the intersection with slice I, and yF at the intersection with slice F. Now think of each slice as a frame in a movie, and run the movie backwards. The result corresponds to an inverted stack of simultaneity slices, with slice F at the bottom and slice I at the top (see Fig. 2). The order of the slices changes, and the particle is now seen to move ‘‘from right to left’’. But—so the argument goes— the configuration of E and B vectors within each slice does not change. In particular, the E and B vectors at each point on the particle worldline are as before. So the angles between them are as before. In the original movie, we see the angle yI at the beginning, and see yF at the end. When the movie is run backwards, the order is reversed. We first see yF and then later yI : (We do not see ðp  yF Þ followed by ðp  yI Þ; as we should if the B vectors had been inverted in each frame before the movie was run backwards.) In this picture, the vector fields E and B in each slice ‘‘just lie there’’ and are kept intact when the order of time slices is reversed. I will eventually try to make the case that the picture is misleading. Roughly speaking, the idea will be that, because B is

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an ‘‘axial’’ vector field, it can no more just lie there than an angular velocity vector field can; and the inversion of B under time reversal is no more mysterious than the change from clockwise rotation to counterclockwise rotation induced by that operation. But first I will switch attention to the four-dimensional invariant formulation of classical electromagnetic theory, and give a direct argument for the time reversal invariance of the theory that makes no reference to magnetic fields at all.

3. The invariant formulation of classical electromagnetic theory In this section, I review the invariant formulation of classical electromagnetic theory within the framework of relativity theory.9 In what follows, let ðM; gab Þ be a relativistic spacetime, i.e., let M be a smooth, connected, four-dimensional manifold, and let gab be a smooth pseudo-Riemannian metric on M with signature ð1; 3Þ: (With this sign convention, a vector xa at a point of M qualifies as timelike if xa xa > 0 and spacelike if xa xa o0: It is null if xa xa ¼ 0:) Furthermore, let ra be the unique (torsion free) derivative operator on M compatible with gab (i.e., ra gbc ¼ 0). Finally, let ta be a continuous timelike vector field on M:10 ta determines a ‘‘temporal orientation’’ on M: I will say that a timelike vector xa at a point is future-directed relative to ta if ta xa > 0 and past-directed relative to ta if ta xa o0: (The first condition just captures the requirement that xa and ta be co-oriented at the point, i.e., that they belong to the same lobe of the null cone, and the second that they belong to opposite lobes.11) In what follows, we will need to keep track of the role played by this temporal orientation in our definitions and constructions. In the invariant formulation of electromagnetic theory, one works with a smooth, anti-symmetric tensor field Fab on M that represents the electromagnetic field itself, and a smooth vector field J a on M that represents the charge-current field that serves as its source. Consider first Fab : We can think of it as coding, for every point in spacetime, the electromagnetic force12 that would be experienced by a point test particle there, depending on its charge and instantaneous velocity.13 But just how it does so is a bit delicate because of a certain ambiguity in the notion of ‘‘instantaneous velocity’’. One must distinguish between senses that do and do not depend on a choice of temporal orientation. 9

I will assume familiarity with the basic elements of relativistic spacetime geometry, and review just a few points of special importance. A good reference for all the material in this section and those that follow, including the abstract index notation that I will be using, is Wald (1984). 10 Thus I am assuming that ðM; gab Þ is temporally orientable, i.e., that there exists such a field on M: (Note that timelike vectors are automatically non-vanishing since the zero vector at every point is null.) 11 Note that all timelike vectors at all points qualify as either future-directed or past-directed relative to ta ; since it is not possible that ta xa ¼ 0: No two timelike vectors can be orthogonal. 12 The force experienced by a charged particle in the presence of an electromagnetic field can be decomposed into separate ‘‘electric’’ and ‘‘magnetic’’ components—we will see how this works later—but here I am considering only the net, undifferentiated force. 13 A ‘‘test particle’’ is one whose own contribution to the electromagnetic field is negligible and may, at least for the purposes at hand, be ignored.

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Fig. 3. The ‘‘instantaneous velocity’’ of a point particle at a point can be represented either as a tangent line, or as a four-velocity vector, there. The latter depends on a choice of temporal orientation. The former does not.

Consider a point particle with non-zero mass.14 We can represent its worldline as the image of a smooth timelike curve in M; and take its tangent line at any point (in the tangent space of the point) to represent its instantaneous velocity there (see Fig. 3). It is a basic fact of electromagnetic life that the net electromagnetic force experienced by the particle at a point depends only on its tangent line L there and its charge q. So we have a force map of form /L; qS/F ðL; qÞ at every point.15 It is crucially important for our purposes that this structural characterization of the electromagnetic field makes no reference to the background temporal orientation. But when it comes to actually specifying the map, it is convenient to allow such reference. One would like to say, for example, that, for given q; the induced map L/F ðL; qÞ is ‘‘linear’’. But it must be explained what that means, since we are dealing here with a map from lines to vectors. The standard way is to redescribe the map as one from vectors to vectors using the background temporal orientation as an auxiliary structure. Rather than representing the particle’s instantaneous velocity at a point as a tangent line, we represent it as a unit timelike vector there (co-aligned with the tangent line). But there are two from which to choose. (See Fig. 3.) One is future-directed with respect to ta ; the other is past-directed. It makes no difference which we choose, but, at least for the moment, let us systematically (i.e., at every point) work with the one that is future-directed. So far, we have recast the force map at every point as one of form /xa ; qS/F ðxa ; qÞ; where xa is a four-velocity vector, i.e., a future-directed, unit timelike vector, there.16 Now we can avail ourselves of the mathematical formalism of linear algebra to recover the standard representation of the electromagnetic field. It is another basic fact of electromagnetic life that the force map under consideration satisfies the following three simple conditions at every point: (i) For all q; the induced map xa /F ðxa ; qÞ is linear. 14

Here and in what follows, ‘‘mass’’ will be understood to mean rest mass. Here I have temporarily dropped tensor indices. I hope this does not cause confusion. I could equally well have written: /L; qS/F a ðL; qÞ: 16 Until further notice, let us agree that all references to temporal orientation, four-velocities, etc. should be understood to be determined relative to ta : 15

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(ii) For all four-velocity vectors xa and all q; the force vector F ðxa ; qÞ is orthogonal to xa : (iii) For all four-velocity vectors xa ; F ðxa ; qÞ is directly proportional to q in the sense that q1 F ðxa ; q1 Þ ¼ F ðxa ; q2 Þ q2 for all q1 and all q2 a0: The three conditions (jointly) are equivalent to the requirement that there exist an anti-symmetric tensor field Fab on M such that, at every point,17 F ðxa ; qÞ ¼ qF ab xb :

ð17Þ

In this way we are led to the field Fab with which we started.18 In our characterization of Fab ; we made essential reference to the background temporal orientation determined by ta : At a crucial stage, we replaced tangent lines by unit tangent vectors, future-directed with respect to ta : Schematically, we have 9 invariant map from > > > tangent lines to force vectors = ) Fab : > þ > > ; temporal orientation But it should be stressed, once again, that the electromagnetic field itself can be fully characterized without reference to temporal orientation (as a map from tangent lines to force vectors). A similar story can be told about the second element in our initial pair ðFab ; J a Þ; the vector field J a on M that represents the charge-current source for the electromagnetic field. It is another basic fact of electromagnetic life that the (scalar) charge density observed at any point by an individual is determined solely by his or her tangent line L there. So we have a map of structure L/JðLÞ at every point. If we 17

q

The argument proceeds in three steps. It follows from (i) that, for each q; there is a tensor field F ab on q

1

M such that, at every point, F ðxa ; qÞ ¼ F ab xb for all four-velocity vectors xa : If we now take Fab to be F ab ; it follows immediately from (iii) that 1 q F ðxa ; qÞ ¼ F ðxa ; 1Þ ¼ q F ab xb ¼ qF ab xb : 1 Now, finally, condition (ii) implies that, at all points, Fab xa xb ¼ ðF ab xb Þxa ¼ 0 for all four-velocity vectors xa : Since the tangent space at a point is spanned by the set of four-velocity vectors there, it follows that Fab ¼ Fba everywhere. (The converse holds too, for if Fab is an anti-symmetric tensor field on M; and Eq. (17) holds, then the force field /xa ; qS/F ðxa ; qÞ certainly satisfies conditions (i)–(iii) at every point.) 18 Note that there can be only one tensor field Fab on M satisfying (17) at all points. For if two fields 1

2

1

2

F ab and F ab code the same force values at all points, i.e., if ðF ab  F ab Þxb ¼ 0 for all four-velocity vectors xa 1

2

at all points, then F ab ¼ F ab : This follows because the set of four-velocity vectors at any point of M spans the tangent space there.

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systematically replace tangent lines L with four-velocity vectors xa ; the map can be recast in the form xa /Jðxa Þ; i.e., as one from vectors to scalars (rather than from lines to scalars). The map, it turns out, is linear. (This is the last basic fact of electromagnetic life I will need to cite for the characterization of Fab and J a :) Linearity implies (and is fully equivalent to the requirement) that there is a vector field J a on M such that, at every point, Jðxa Þ ¼ J a xa

ð18Þ

for all four-velocity vectors xa :19 So in this case, schematically, we have 9 invariant map from > > > tangent lines to charge density values = ) Ja: > þ > > ; temporal orientation With the fields ðFab ; J a Þ in hand, we can formulate the most basic laws of classical electromagnetic theory: (i) the (Lorentz) equation of motion for a charged test particle in the presence of an electromagnetic field, and (ii) Maxwell’s field equations. Once again, we can represent the world line of a test particle as the image of a smooth timelike curve of form g : I-M; where I is a connected subset of R: For convenience, we assume that the curve is so parametrized that the tangent vector xa to the curve at every point is of unit length and future-directed. Then xa is called the four-velocity of the particle at the point, and the directional derivative xb rb xa there is called its four-acceleration. All the ingredients are now assembled to insert into ‘‘F ¼ ma’’. Suppose the particle has mass ma0 and charge q: Then at every point, the force acting on it is qF ab xb : So its equation of motion comes out as: qF ab xb ¼ mxb rb xa :

ð19Þ

Note that the particle’s four-acceleration at every point is independent of the background temporal orientation. (If we reparametrize g so that the tangent vector to the curve at every point is of unit length and past-directed, i.e., so that it is xa rather than xa ; then its four-acceleration is ðxb Þrb ðxa Þ ¼ xb rb xa :) In our notation, Maxwell’s equations come out as r½a Fbc ¼ 0;

ð20Þ

rn F na ¼ J a :

ð21Þ

It is not my purpose to develop electromagnetic theory, but it is worth mentioning two immediate consequences of these equations. (i) The first asserts that Fab is closed. So, by the converse to the Poincare! lemma, it is, at least locally, exact, i.e., given any point p in M; there is an open set O containing p; and a smooth field Aa on 19

Clearly, there can be only one such field. The argument is much the same as given above for the 1

2

1

2

uniqueness of Fab : If ðJ a  J a Þxa ¼ 0 for all four-velocity vectors xa at all points, then J a ¼ J a :

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O; such that Fab ¼ r½a Ab : (ii) The second equation implies that ra J a ¼ 0; which captures the requirement of local charge conservation.20 4. Time reversal invariance As already indicated, the objects Fab and J a used to represent the electromagnetic field and its charge-current source field are only defined relative to a background choice of temporal orientation. The time reversal operation is naturally understood as one taking fields on M as determined relative to one temporal orientation to corresponding fields on M as determined relative to the other. In the case at hand, we thus have an operation of form ðM; gab ; ta ; Fab ; J a Þ

time reversal

/

ðM; gab ; ta ; T Fab ; T J a Þ

to consider.21 Here T Fab and T J a are understood to be the unique tensor fields on M that bear the same relation to ta that Fab and J a bear to ta : I claim first that T

Fab ¼ Fab ;

T a

J ¼ J a :

ð22Þ ð23Þ

The arguments for the two cases are very much the same. Recall, first, our route from the force map /L; qS/F ðL; qÞ to the tensor field Fab : At a crucial stage we traded in tangent lines for unit vectors. Here is a complete statement of the condition that characterizes Fab : for all points p; all charge values q; and all timelike lines L in the tangent space at p; the force F ðL; qÞ on a test particle at p with charge q and (invariant) instantaneous velocity L is given by F ðL; qÞ ¼ qF ab xb ;

ð24Þ

a

where x is the unit timelike vector at p; co-alligned with L; that is future directed with respect to ta : We need to consider what happens if we keep this characterization 20

Since F ab is anti-symmetric, we have ra J a ¼ ra rn F na ¼ r½a rn F na : But if Rabcd is the Riemann curvature field associated with rn ; and Rbc ¼ Rabca is the associated Ricci curvature field, we also have 2r½a rn F na ¼  Rnman F ma  Rnman F nm ¼ Rnman F ma þ Rnman F nm ¼  Rma F ma þ Rmn F nm ¼ Rmn F mn þ Rnm F mn ¼  Rmn F mn þ Rmn F mn ¼ 0: All the component assertions here follow from basic facts about Rabcd and Rbc : (See Wald (1984, Section 3.2). Warning: the sign conventions in that book differ slightly from mine.) The first equality holds for any tensor field F ab ; anti-symmetry plays no role. The second follows from the fact that Rabcd ¼ 0: The third and fourth, respectively, involve nothing more than the definition of Rab and systematic index substitution. The fifth equality follows from the symmetry of Rmn : 21 I hope my imperfect notation does not cause confusion. I am using the same symbol ‘T’ for the time reversal operator that I used in Section 2, even though the operator is understood somewhat differently here. In a sense that can be made precise, the two ways of understanding it are equivalent if the background relativistic spacetime admits a ‘‘frame of reference’’ (see Section 6). In particular, they are equivalent in Minkowski spacetime.

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intact, but replace ta in the italicized expression by ta ; i.e., if we represent the tangent line L; not by the unit vector xa ; but rather by xa :22 In that case we are led to an anti-symmetric tensor field T Fab on M that satisfies an equation parallel in form to (24), namely F ðL; qÞ ¼ qðT F ab Þ ðxb Þ:

ð25Þ

T

To determine Fab ; we need only compare (25) with (24). It follows from the two conditions that, at all points, ðT F ab þ F ab Þxb ¼ 0; for all unit timelike vectors xa future-directed with respect to ta :23 This implies that T Fab ¼ Fab (since the set of all such vectors at any point spans the tangent space there). Consider next the corresponding argument for (23). The complete statement of the condition characterizing J a is this: for all points p; and all timelike lines L in the tangent space at p; the charge density determined by an observer at p with (invariant) instantaneous velocity L is given by ð26Þ JðLÞ ¼ J a xa ; where xa is the unit timelike vector at p; co-alligned with L; that is future directed with respect to ta . If we replace ta by ta in the italicized expression, we are led to a field T a J on M satisfying an equation parallel in form to (26), namely24 JðLÞ ¼ ðT J a Þðxa Þ: ð27Þ It then follows from (26) and (27) that ðT J a þ J a Þxa ¼ 0; for all unit timelike vectors xa future-directed with respect to ta : And this, in turn, implies that T J a ¼ J a : Suppose now we agree that T Fab ¼ Fab and T J a ¼ J a : Then it follows immediately that Maxwell’s equations are time reversal invariant. (That is, the initial fields ðFab ; J a Þ satisfy the equations iff the time reversed fields ðT Fab ; T J a Þ do.) r½a Fbc ¼ 0 3 r½a ðFbc Þ ¼ 0 3 r½a ðT Fbc Þ ¼ 0; rn F na ¼ J a 3 rn ðF na Þ ¼ J a 3 rn ðT F na Þ ¼ T J a : The computations here are trivial. But it is an important, non-trivial property of Maxwell’s equations that they are time reversal invariant. This outcome was not guaranteed by our characterization of Fab ; J a ; and the time reversal operator T. Consider, for example, the following alternative (totally unmotivated) field equation: ra ðFmn F mn Þ ¼ Ja :

ð28Þ

Clearly, this one is not time reversal invariant. The left side tensor field is invariant under time reversal: ra ððT Fmn Þ ðT F mn ÞÞ ¼ ra ðFmn F mn Þ: But the right side field is not: ðT Ja Þ ¼ Ja aJa : The difference between (28) and the second Maxwell equation 22 I take for granted that the charge (and mass) of a particle are not dependent on temporal orientation. One could challenge this assumption, perhaps, but Albert shows no inclination to do so. His concerns are entirely different, and cannot even be reached unless this much is assumed. 23 Of course, we could just as well formulate this as the condition that ðT F ab þ F ab ÞZb ¼ 0 for all unit timelike vectors Za future-directed with respect to ta : 24 Here I am assuming that charge density at a point, as determined by an observer, does not depend on temporal orientation.

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ðrn F na ¼ J a Þ just comes to this. In both cases, the tensor fields asserted to be equal depend on (or are defined relative to) the background temporal orientation. But only in the second case do the dependencies match.

5. Decomposition of the electromagnetic field I have argued that classical electromagnetic theory is time reversal invariant without making any reference to component electric and magnetic fields whatsoever. I turn to the latter in this section. So far, we have under consideration a relativistic spacetime ðM; gab Þ; a continuous timelike vector field ta on M; and a pair of smooth fields ðFab ; J a Þ representing the electromagnetic field and its charge-current source field (relative to the temporal orientation determined by ta ). Further background geometric structure must be introduced if we are to decompose Fab and J a : (It is only relative to that additional structure that the decomposition is defined.) A volume element on ðM; gab Þ is a smooth tensor field eabcd on M that is completely anti-symmetric ðe½abcd ¼ eabcd Þ and satisfies the normalization condition eabcd eabcd ¼ 24:25 If there exists a volume element on ðM; gab Þ; it is said to be orientable. In this case, there exist exactly two volume elements on ðM; gab Þ altogether. If eabcd is one, the other is eabcd : In what follows, I will assume that our background spacetime ðM; gab Þ is orientable (as well as temporally orientable), and take eabcd to be a volume element on ðM; gab Þ: I will also assume that ðM; gab Þ admits what I will call a ‘‘frame of reference’’, namely a smooth, future-directed, unit timelike vector field Za on M that is constant (i.e., ra Zb ¼ 0).26 Let Za be one such in what follows. It determines, at least locally, a foliation of spacetime into a family S of spacelike hypersurfaces that are everywhere orthogonal to Za :27 These hypersurfaces will play the role of the simultaneity slices taken for granted in Section 2. We can now recover the standard textbook formulation of classical electromagnetic theory by decomposing Fab and J a relative to Z and eabcd : Consider the 1

25

2

3

4

1

2

3

4

Given an ordered set of vectors ðxa ; xa ; xa ; xa Þ at a point, we can think of eabcd xa xb xc xd as the ‘‘oriented volume’’ of the parallelepiped determined by the vectors. Orientation enters in that the order of 2

1

3

4

1

2

3

4

the vectors in the quadruple matters, e.g., eabcd xa xb xc xd ¼ eabcd xa xb xc xd : The normalization condition eabcd eabcd ¼ 24 is equivalent to the requirement that, for all orthonormal bases 1

2

3

4

1

2

3

4

fxa ; xa ; xa ; xa g at all points, eabcd xa xb xc xd ¼ 71: 26 The condition of constancy is more stringent than necessary here, but it will allow us to sidestep certain complexities that are of no special importance for our purposes. Indeed, it would be no great loss if we restricted attention in this section to Minkowski spacetime (which certainly admits ‘‘frames of reference’’). 27 If Za is constant, it is closed ðr½a Zb ¼ 0Þ: So, at least locally, it is exact, i.e., there exists a smooth scalar field t such that Za ¼ ra t: The t=constant hypersurfaces are spacelike, and everywhere orthogonal to Za :

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following fields on M: hab ¼ gab  Za Zb ;

ð29Þ

r ¼ J a Za ;

ð30Þ

j a ¼ hab J b ;

ð31Þ

E a ¼ F ab Zb ;

ð32Þ

1 Ba ¼ eabcd Zb Fcd : 2

ð33Þ

Let us consider them in turn. We can think of hab ; first, as a projection field that, at every point, projects vectors onto their components orthogonal to Za : To see this, consider a vector la at some point. We can express it in the form: la ¼ ðlb Zb ÞZa þ ðgab  Za Zb Þ lb : The first term on the right is proportional to Za : The second is orthogonal to Za (since Za Za ¼ 1 and, therefore, ððgab  Za Zb Þ lb ÞZa ¼ ðla Za Þ  ðZb lb Þ ¼ 0). In particular (taking J a for la ), we have J a ¼ ðJ b Zb ÞZa þ hab J b ; i.e., J a ¼ rZa þ j a :

ð34Þ

r is the charge-density relative to Za ; and j a is the current-density relative to Za : (We have already encountered the former in our characterization of J a :) Since j a is orthogonal to Za at every point, it is everywhere tangent to the hypersurfaces in S: So we can think of j a as ‘‘residing’’ on them. (This will be true of E a and Ba as well.) Note that ra hab ¼ 0; since ra gbc ¼ 0 ¼ ra Zb : Next, E a is the electric field as determined relative to Za : It is orthogonal to Za at every point (since Fab is anti-symmetric). It follows from our characterization of Fab that E a is also the undifferentiated, net electromagnetic force that would be felt by a test particle of unit positive charge with four-velocity Za : This can be captured as the standard claim that, as determined relative to any particular frame of reference, a test particle at rest in the frame experiences an electric force, but no magnetic force. Ba is the magnetic field as determined relative to Za and eabcd : The latter dependency is crucial. It too is orthogonal to Za at every point (since eabcd is completely antisymmetric and so eabcd Za Zb ¼ 0). Thus, E a and Ba are tangent everywhere to the hypersurfaces in S: It follows from (29)–(33), with just a bit of computation, that Fab ¼ 2Z½a Eb þ eabcd Zc Bd :

ð35Þ

Again, it is not my purpose to develop electromagnetic theory, but it is worth mentioning two consequences of (35): F ab Fab ¼ 2ðE a Ea  Ba Ba Þ; eabcd F ab F cd ¼ 8E a Ba :

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They establish that ðE a Ea  Ba Ba Þ and E a Ba are invariant, i.e., have the same value for all choices of the background frame of reference Za : (In three-vector notation, they would be expressed as ðjBj2  jEj2 Þ and E  B:28) It also follows from (35) and (29)–(33) that we have the following equivalences: ( ra Ba ¼ 0; r½a Fbc ¼ 0 3 eanbc Zn rb Ec ¼ Zn rn Ba ; ( rn F

na

¼J 3

ra E a ¼ r;

a

eanbc Zn rb Bc ¼ Zn rn E a þ j a :

Each of the equations on the right captures, in our notation, one of the four Maxwell equations, resp. (10), (9), (8), and (11).

6. Time reversal invariance once again We need to determine how the time reversal operator T acts on r; j a ; E a ; and Ba : It will help to first consider, side by side, three senses of orientation: temporal, spatial, and spatiotemporal. The first and third are determined directly by ta and eabcd ; respectively. We have encountered them already. A timelike vector xa at a point is future directed or past directed relative to ta ; of course, depending on whether ta xa is positive or negative. 1

2

3

4

Similarly, an ordered set of linearly independent vectors ðxa ; xa ; xa ; xa Þ at a point has positive spatiotemporal orientation or negative spatiotemporal orientation relative 1

2

3

4

to eabcd depending on whether its oriented volume eabcd xa xb xc xd is positive or negative.29 The notion of spatial orientation is slightly more delicate, because it presupposes a foliation of spacetime into a family of spacelike hypersurfaces (that represent ‘‘space’’ at a given ‘‘time’’). Once again, let Za be a frame of reference on ðM; gab Þ; and let S be the associated family of spacelike hypersurfaces everywhere orthogonal to Za : We may take a spatial volume element relative to Za to be a smooth field eabc that is completely antisymmetric ðe½abc ¼ eabc Þ; normalized so that eabc eabc ¼ 6; and orthogonal to Za ; i.e., eabc Za ¼ 0:30 We can think of eabc as residing on the hypersurfaces in S: An ordered 1

2

3

set of linearly independent vectors ðsa ; sa ; sa Þ at a point, all orthogonal to Za ; has 28

The sign changes arise because E a and Ba are spacelike. 1

2

3

4

1

2

3

4

It cannot be 0 since eabcd xa xb xc xd ¼ 0 iff the set fxa ; xa ; xa ; xa g is linearly dependent. 30 Clearly, it follows from anti-symmetry that if eabc Za ¼ 0; then also eabc Zb ¼ eabc Zc ¼ 0: The normalization condition eabc eabc ¼ 6 is equivalent to the requirement that, for all orthonormal bases 29

1

2

3

1

2

3

of form fZa ; sa ; sa ; sa g; at all points, eabc sa sb sc ¼ 71:

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positive spatial orientation or negative spatial orientation relative to eabc depending on 1

2

3

whether eabc sa sb sc is positive or negative. There are exactly two spatial volume elements relative to Za on ðM; gab Þ; namely eanbc Zn and eanbc Zn :31 I have construed time reversal as an operation taking fields on M as determined relative to ta to corresponding fields as determined relative to ta : We can understand ‘‘spatiotemporal parity reversal’’ and ‘‘spatial parity reversal’’ (with ‘‘space’’ taken relative to Za ) similarly. Specifically, we can take the former to be an operation taking fields as determined relative to eabcd to fields as determined relative to eabcd : And we can take the latter to be one taking fields as determined relative to eabc ¼ eanbc Zn to ones as determined relative to eabc ¼ eanbc Zn : The three operations under consideration are not independent of each other. The time reversal operation takes Za to Za : So, for example, a combination of time reversal and spatiotemporal parity reversal ðeabcd /  eabcd Þ leaves spatial orientation intact: eabc ¼ eanbc Zn /ðeanbc ÞðZn Þ ¼ eabc : There are three composite possibilities in all. T time reversal without spatial parity reversal :

ta /  ta ; eabcd /  eabcd ; eabc /eabc :

P spatial parity reversal without time reversal :

ta /ta ; eabcd /  eabcd ; eabc /  eabc :

TP time reversal and spatial parity reversal :

ta /  ta ; eabcd /eabcd ; eabc /  eabc :

Let us consider how all three operations T; P; and TP act on Fab ; J a ; and the fields that figure in their decomposition. The full set comes out as in Table 1. The entries in the third column are determined by the those in the first two, since TP is the product, or composition, of T and P (in either order).32 The entries in the second column are all straightforward except for the final one. None of the listed fields other than

31 The choice for the index of contraction does not matter. I use the second index to match the choice in Section 3. 32 So, for example, TP Ba ¼ T ðP Ba Þ ¼ T ðBa Þ ¼ T ðT Ba Þ ¼ Ba : (The final step follows from the modest assumption that the time reversal operation is self-cancelling in the sense that T2 ¼ I:)

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Table 1

The action of the operations T; P; and TP on Fab ; J a ; and the fields that figure in their decomposition. The left box shows the standard time reversal transformations. The right box shows the ‘‘Albert alternative transformations’’.

Ba ¼ 12 eabcd Zb Fcd ¼ 12 eacd Fcd depends on eabc ; so none is affected by the operation of parity reversal. For Ba itself, we have33 P

Ba ¼ 12 ðP eacd ÞðP Fcd Þ ¼ 12 ðeacd ÞFcd ¼ Ba :

That leaves only the entries in the first column to check. We have already argued that Fab ¼ Fab (22) and T J a ¼ J a (23). The time reversal operation flips future-directed timelike vectors to past-directed ones. So we have T Za ¼ Za and, therefore, T hab ¼ T a ðg b  Za Zb Þ ¼ ½ðT gab Þ  ðT Za ÞðT Zb Þ ¼ ½gab  ðZa ÞðZb Þ ¼ hab : The final four entries in the first column are easy to check as well. T

T

r ¼ ðT J a ÞðT Za Þ ¼ ðJ a ÞðZa Þ ¼ r;

T a

j ¼ ðT hab Þ ðT J b Þ ¼ hab ðJ b Þ ¼ j a ;

T

E a ¼ ðT F ab Þ ðT Zb Þ ¼ ðF ab ÞðZb Þ ¼ E a ;

T

Ba ¼ 12 ðT eacd ÞðT Fcd Þ ¼ 12 eacd ðFcd Þ ¼ Ba :

I have bothered to produce the table because it helps to clarify the relation of the standard time reversal transformations (4)–(7) to David Albert’s proposed 33

Note that T and P (and, so, also TP) commute with the operations of raising and lowering indices. This follows since neither gab nor its inverse gab depends on the background temporal or spatial orientation, and therefore T gab ¼ gab ¼ P gab and T gab ¼ gab ¼ P gab : So, for example, P acd

e

¼ P ðgam gcn gdp emnp Þ ¼ ðP gam ÞðP gcn ÞðP gdp ÞðP emnp Þ ¼ gam gcn gdp ðemnp Þ ¼ eacd :

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alternatives (13)–(16). They correspond, respectively, to the final four entries in the T and TP columns. This really is the heart of the matter. Albert’s transformations should properly be seen as characterizing a composite operation of time and parity reversal. Magnetic fields do not ‘‘just lie there’’ (in the sense discussed in Section 2) under time reversal. But they are left intact under TP because both component operations invert them (and the two actions cancel each other). I should anticipate one possible point of confusion. It was asserted in Section 2 that Maxwell’s equations are not (all) invariant under Albert’s alternative transformations. Now it is claimed that those transformations characterize the TP operation. Does it not follow that Maxwell’s equations are not invariant under the composite operation of time and parity reversal? (And, if so, doesn’t that conclusion fly in the face of orthodoxy just as much as Albert’s claim that they are not invariant under time reversal alone?) The answer to the first question is ‘no’. Here is why in the language of Section 2. Consider, for example, Eq. (9): r  E ¼ @B=@t: It is not invariant under Albert’s transformations if they are understood as time reversal transformations. For in that case, E and B are left intact, and the partial derivative operator @=@t is replaced by @=@ðtÞ ¼ @=@t: But (9) is invariant under Albert’s transformations if they are understood as combined time and parity reversal transformations. In the latter case, cross-products and curls (such as r  E) are also multiplied by 1; and that restores the balance between the two sides of the equation. The transformation properties for Ba are exactly the same as for angular velocity. Let xa be a (not necessarily constant) future-directed, unit timelike vector field on M: We may think of xa as the four-velocity field of a fluid. The angular velocity field of the fluid is then given by oa ¼ 12 eabcd xb rc xd : Applying the T; P; and TP operations to oa yields T

oa ¼ 12 ðT eabcd ÞðT xb Þrc ðT xd Þ ¼ 12 ðeabcd Þðxb Þrc ðxd Þ ¼ oa ;

P

oa ¼ 12 ðP eabcd ÞðP xb Þrc ðP xd Þ ¼ 12 ðeabcd Þxb rc xd ¼ oa ;

TP

oa ¼ 12 ðTP eabcd ÞðTP xb Þrc ðTP xd Þ ¼ 12 eabcd ðxb Þrc ðxd Þ ¼ oa :

These correspond to the entries for ðT Ba Þ; ðP Ba Þ; and ðTP Ba Þ in the final row of the table.34 Here we have worked with what might be called the ‘‘invariant angular velocity’’ of xa : If we worked, instead, with the angular velocity of xa as determined relative to the frame Za ; the formal parallel with Ba would be even closer. The later ‘‘relativized angular velocity’’ is defined by 34

# a ¼ 12 eacd rc xd ¼ 12 eabcd Zb rc xd : o Here the dependence on eabc ; characteristic of axial vector fields, is explicit. (To see where this expression a comes from, notice that if x# is hab xb ; the field that results from projecting xa orthogonal to Za ; then the curl a of x# (relative to Za ) is 1 eacd rc x# d : But, since ha is constant, and eacd Z ¼ eabcd Z Z ¼ 0; 2

1 acd rc x# d 2e a

¼

1 acd a h b rc xn ¼ 12 eacd ðgnd 2e a a

d

b

n

 Zd Z Þrc xn ¼

b

d

1 acd rc xd : 2e

# transforms exactly like oa under the operations T; P; and TP: # is just the curl of x# :) o So o

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Think about it this way. If we make a movie of a fluid whirling in a clockwise direction, and then play the movie backwards, we see the fluid whirling in a counterclockwise direction. The angular velocity of the fluid is reversed. On the other hand, if we play it backwards, project the image onto a mirror, and then watch the reflected image, we see the fluid whirling in a clockwise direction again, as in the original. In this case, the angular velocity is not reversed. I can imagine David Albert objecting that the proposed analogy between a magnetic field and an angular velocity field is misleading. Presumably, he would agree that the latter (understood as a component of a fluid’s instantaneous dynamical condition) is properly reversed by the time reversal operation, just as is the linear velocity of a point particle. (Recall the discussion in Section 2.) But he would continue to insist that ‘‘magnetic fields are not—either logically or conceptually—the rates of change of anything’’, and so should not be reversed. Of course, I do not think the response is to the point. Magnetic fields may not be rates of change of anything in the appropriate sense, but they are axial vector fields. The claim here is that one does not have a magnetic field Ba until a spatial orientation is specified (in addition to a temporal orientation). And, similarly, one does not have a time reversed field ðT Ba Þ until a spatial orientation is specified (in addition to the reversed temporal orientation). At issue is whether the second spatial orientation is taken to be the same as the first or not i.e., whether we are considering the action of T on Ba or, rather, TP: The first option leads to the standard account of how the time reversal operator acts on magnetic fields; the second leads to Albert’s account. One wants to say to him that the time reversal operation, properly understood, simply does not—‘‘either logically or conceptually’’—include spatial parity reversal as a constituent sub-operation. 7. Conclusion I have tried to show that David Albert’s claims concerning the (non) time reversal invariance of classical electromagnetic theory are based on a misleading way of thinking about the representation of electromagnetic fields, and that some insight is achieved into issues of time reversal invariance when one thinks about them in terms of four-dimensional spacetime geometry. I have argued (Section 6) that the inversion of magnetic fields under time reversal that so troubles Albert is benign and makes good geometric sense. I have also argued (Section 4) that one can formulate and defend the claim that classical electromagnetic theory is time reversal invariant without making reference to magnetic fields at all. Defending orthodoxy is much less exciting work than embracing heresy. But it has its place on occasion. I hope I have succeeded in restoring the faith of at least a few of my colleagues. Acknowledgements I wish to thank Frank Arntzenius, Jordi Cat, Erik Curiel, Dennis Dieks, Robert Geroch, Tim Maudlin, John Norton, Howard Stein, Jos Uffink, and an anonymous

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referee for helpful comments. (They certainly do not all agree with the claims I have made.)

References Albert, D. Z. (2000). Time and chance. Cambridge, MA, London: Harvard University Press. Arntzenius, F. (2000). Are there really instantaneous velocities? The Monist, 83, 187–208. Arntzenius, F. (2004). Time reversal operations and the direction of time. Studies in the History and Philosophy of Modern Physics, 35(1), 31–44. Earman, J. (2002). What time reversal invariance is and why it matters. International Studies in the Philosophy of Science, 16, 245–264. Floyd, S. (2003). On the possibility of instantaneous velocities. unpublished. Smith, S. (2003). Are instantaneous velocities real and really instantaneous?: An argument for the affirmative. Studies in the History and Philosophy of Modern Physics, 34, 261–280. Wald, R. M. (1984). General relativity. Chicago: University of Chicago Press.